我們首先回顧複幾何基本的背景知識，接著介紹關於卡拉比-丘流形的經典結果，比如形變，BTT定理，凱勒錐。進而我們研究複模空間以及凱勒模空間上的一些基本結構，並把成對的鏡像流形相應的模空間上的訊息對等起來，給出一般性的鏡像對稱猜想的敘述。最終，我們將這一想法應用到由五次式所給出的卡拉比-丘三維流形上。;Mirror symmetry is a mysterious relationship between pairs of Calabi-Yau manifolds $X$ and $\check{X}$, arising from string theory. Roughly speaking, it exchanges things related to the complex structure of $X$ with things related to the symplectic structure of $\check{X}$, and vice versa. In this article, we will give an introduction to the tools needed to understand the elementary mirror symmetry conjecture. It will end with a detailed working out of the example of the quintic.
We begin with the necessary background on complex geometry, and then introduce some of the classical results about Calabi-Yau manifold including deformation, BTT theorem and K\"ahler cone. Next we study some essential structures on the complex moduli space and the K\"ahler moduli space. We equate the data of structures arising on these two sides for mirror pairs of Calabi-Yau manifolds, and state the general mirror symmetry conjecture, and finally using this idea carry out the example of quintic Calabi-Yau 3-fold.
